算法的增长函数？

Question

Well i have two questions here:-

1. If f(n) is function whose growth rate is to be found then, Is for all three notations will the g(n) be same, like for f(n)=O(g(n)) and similaraly for omega and theta ?

2. Theta notation is "omega and Oh" if in some case if oh and omega functions are different then, how will we find theta function there ? Thanks :)

Answer 1

O, Θ and Ω notation represent related but very different concepts. O-notation expresses an asymptotic upper bound on the growth rate of a function; it says that the function is eventually bounded from above by some constant multiple of some other function. Ω notation is similar, but gives a lower bound. Θ notation gives an asymptotic tight bound - for sufficiently large inputs, the algorithm grows at a rate that is bounded from both above and below by a constant multiple of a function.

If f(n) = O(g(n)), it is not necessarily true that f(n) = Ω(g(n)) or that f(n) = Θ(g(n)). For example, 1 = O(n), but 1 ≠ Ω(n) because n grows strictly faster than 1.

If you find that f(n) = O(g(n)) and Ω(h(n)), where g(n) ≠ h(n), you may have to do a more precise analysis to determine a function j(n) such that f(n) = Θ(j(n)). If g(n) = Θ(h(n)), then you can conclude that f(n) = Θ(g(n)), but if the upper and lower bounds are different there is no mechanical way to determine the Θ growth rate of the function.

Hope this helps!

Answer 2

f(n)=O(g(n)) means that n>N => |f(n)|≤C|g(n)| for some constants N and C.

f(n)=Ω(g(n)) means that n>N => |f(n)|≥C|g(n)| for some constants N and C.

f(n)=Θ(g(n)) means that f(n)=O(g(n)) and f(n)=Ω(g(n)).

It is not possible for all f to find ag such that f(n)=Θ(g(n)) if we want g to be a "good" function (ie something like n^r*Log(n)^s). For instance, if f(n)=cos(n)²*n+sin(n)²*n², we have f(n)=O(n²) and f(n)=Ω(n) but we can't find a "good" g such that f(n)=Θ(g(n)).